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Mirrors > Home > MPE Home > Th. List > dm0rn0 | Structured version Visualization version GIF version |
Description: An empty domain is equivalent to an empty range. (Contributed by NM, 21-May-1998.) |
Ref | Expression |
---|---|
dm0rn0 | ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | alnex 1744 | . . . . . 6 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑥∃𝑦 𝑥𝐴𝑦) | |
2 | excom 2096 | . . . . . 6 ⊢ (∃𝑥∃𝑦 𝑥𝐴𝑦 ↔ ∃𝑦∃𝑥 𝑥𝐴𝑦) | |
3 | 1, 2 | xchbinx 326 | . . . . 5 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ¬ ∃𝑦∃𝑥 𝑥𝐴𝑦) |
4 | alnex 1744 | . . . . 5 ⊢ (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ¬ ∃𝑦∃𝑥 𝑥𝐴𝑦) | |
5 | 3, 4 | bitr4i 270 | . . . 4 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦) |
6 | noel 4178 | . . . . . 6 ⊢ ¬ 𝑥 ∈ ∅ | |
7 | 6 | nbn 365 | . . . . 5 ⊢ (¬ ∃𝑦 𝑥𝐴𝑦 ↔ (∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅)) |
8 | 7 | albii 1782 | . . . 4 ⊢ (∀𝑥 ¬ ∃𝑦 𝑥𝐴𝑦 ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅)) |
9 | noel 4178 | . . . . . 6 ⊢ ¬ 𝑦 ∈ ∅ | |
10 | 9 | nbn 365 | . . . . 5 ⊢ (¬ ∃𝑥 𝑥𝐴𝑦 ↔ (∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅)) |
11 | 10 | albii 1782 | . . . 4 ⊢ (∀𝑦 ¬ ∃𝑥 𝑥𝐴𝑦 ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅)) |
12 | 5, 8, 11 | 3bitr3i 293 | . . 3 ⊢ (∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅) ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅)) |
13 | abeq1 2892 | . . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ ∀𝑥(∃𝑦 𝑥𝐴𝑦 ↔ 𝑥 ∈ ∅)) | |
14 | abeq1 2892 | . . 3 ⊢ ({𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅ ↔ ∀𝑦(∃𝑥 𝑥𝐴𝑦 ↔ 𝑦 ∈ ∅)) | |
15 | 12, 13, 14 | 3bitr4i 295 | . 2 ⊢ ({𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅) |
16 | df-dm 5410 | . . 3 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} | |
17 | 16 | eqeq1i 2777 | . 2 ⊢ (dom 𝐴 = ∅ ↔ {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦} = ∅) |
18 | dfrn2 5602 | . . 3 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} | |
19 | 18 | eqeq1i 2777 | . 2 ⊢ (ran 𝐴 = ∅ ↔ {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦} = ∅) |
20 | 15, 17, 19 | 3bitr4i 295 | 1 ⊢ (dom 𝐴 = ∅ ↔ ran 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 198 ∀wal 1505 = wceq 1507 ∃wex 1742 ∈ wcel 2048 {cab 2753 ∅c0 4173 class class class wbr 4923 dom cdm 5400 ran crn 5401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pr 5180 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-rab 3091 df-v 3411 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-br 4924 df-opab 4986 df-cnv 5408 df-dm 5410 df-rn 5411 |
This theorem is referenced by: rn0 5669 relrn0 5675 imadisj 5782 rnsnn0 5898 f00 6384 f0rn0 6387 2nd0 7501 iinon 7774 onoviun 7777 onnseq 7778 map0b 8238 fodomfib 8585 intrnfi 8667 wdomtr 8826 noinfep 8909 wemapwe 8946 fin23lem31 9555 fin23lem40 9563 isf34lem7 9591 isf34lem6 9592 ttukeylem6 9726 fodomb 9738 rpnnen1lem4 12187 rpnnen1lem5 12188 fseqsupcl 13153 fseqsupubi 13154 dmtrclfv 14229 ruclem11 15443 prmreclem6 16103 0ram 16202 0ram2 16203 0ramcl 16205 gsumval2 17738 ghmrn 18132 gexex 18719 gsumval3 18771 subdrgint 19294 iinopn 21204 hauscmplem 21708 fbasrn 22186 alexsublem 22346 evth 23256 minveclem1 23720 minveclem3b 23724 ovollb2 23783 ovolunlem1a 23790 ovolunlem1 23791 ovoliunlem1 23796 ovoliun2 23800 ioombl1lem4 23855 uniioombllem1 23875 uniioombllem2 23877 uniioombllem6 23882 mbfsup 23958 mbfinf 23959 mbflimsup 23960 itg1climres 24008 itg2monolem1 24044 itg2mono 24047 itg2i1fseq2 24050 itg2cnlem1 24055 minvecolem1 28419 rge0scvg 30793 esumpcvgval 30938 cvmsss2 32066 fin2so 34268 ptrecube 34281 heicant 34316 isbnd3 34452 totbndbnd 34457 rnnonrel 39258 rnmpt0 40854 stoweidlem35 41697 hoicvr 42207 |
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